2025/05/19 by Chun, Carolyn, Douthitt, James Dylan, Ge, Wayne +3
#05B35 #05C15 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.13781
We formulate a geometric version of the Erdős-Hajnal conjecture that applies to finite projective geometries rather than graphs, in both its usual 'induced' form and the multicoloured form. The multicoloured conjecture states, roughly, that a colouring c of the points of PG(n-1,q) containing no copy of a fixed colouring c0 of PG(k-1,q) for small k must contain a subspace of dimension polynomial in n that avoids some colour. If (k,q) = (2,2), then c0 is a colouring of a three-element 'triangle', and there are three essentially different cases, all of which we resolve. We derive both the cases where c0 assigns the same colour to two different elements from a recent breakthrough result in additive combinatorics due to Kelley and Meka. We handle the case that c0 is a 'rainbow' colouring by proving that rainbow-triangle-free colourings of projective geometries are exactly those that admit a certain decomposition into two-coloured pieces. This is closely analogous to a theorem of Gallai on rainbow-triangle-free coloured complete graphs. We also show that existing structure theorems resolve certain two-coloured cases where (k,q) = (2,3), and (k,q) = (3,2).